Suppose X1, X2, …, Xn is a random sample from a distribution...

Suppose $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a distribution with pdf $f(x ; \theta)=(1 / 2) \theta^{3} x^{2} e^{-\theta x}, 0<x<\infty$, zero elsewhere, where $0<\theta<\infty:$ (a) Find the mle, $\hat{\theta}$, of $\theta .$ Is $\hat{\theta}$ unbiased? Hint: Find the pdf of $Y=\sum_{1}^{n} X_{i}$ and then compute $E(\hat{\theta})$. (b) Argue that $Y$ is a complete sufficient statistic for $\theta$. (c) Find the MVUE of $\theta$. (d) Show that $X_{1} / Y$ and $Y$ are independent. (e) What is the distribution of $X_{1} / Y ?$

Suppose $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a distribution with pdf $f(x ; \theta)=(1 / 2) \theta^{3} x^{2} e^{-\theta x}, 0<x<\infty$, zero elsewhere, where $0<\theta<\infty:$
(a) Find the mle, $\hat{\theta}$, of $\theta .$ Is $\hat{\theta}$ unbiased? Hint: Find the pdf of $Y=\sum_{1}^{n} X_{i}$ and then compute $E(\hat{\theta})$.
(b) Argue that $Y$ is a complete sufficient statistic for $\theta$.
(c) Find the MVUE of $\theta$.
(d) Show that $X_{1} / Y$ and $Y$ are independent.
(e) What is the distribution of $X_{1} / Y ?$

Key Concepts

Beta Distribution in the Context of Ratio of Random Variables

The beta distribution often arises as the distribution of a ratio of two gamma distributed random variables, particularly the ratio of one component to the sum of components in a sample. This result is useful in statistical inference for understanding the behavior of proportions and ratios. The identification of a beta distribution in such contexts assists in the derivation of confidence intervals and hypothesis tests related to the parameter of interest.

Sum of Random Variables and the Gamma Distribution

When dealing with independent and identically distributed random variables that follow certain distributions, the sum often follows a gamma distribution. Recognizing the form of the sum’s distribution is crucial because it facilitates the derivation of properties such as its expectation and variance, which are used to assess the properties of estimators. This concept also plays a key role when verifying sufficiency and completeness for the underlying parameter.

Minimum Variance Unbiased Estimator (MVUE)

The minimum variance unbiased estimator (MVUE) is the unbiased estimator with the smallest variance among all unbiased estimators for a given parameter. The MVUE is often obtained by applying the Lehmann–Scheffé theorem, which states that any unbiased estimator that is a function of a complete sufficient statistic is the unique MVUE. This concept is essential in ensuring both unbiasedness and efficiency in parameter estimation.

Transformation and Independence of Statistics

Transformations of random variables, especially when considering ratios or scaling, are essential tools in statistical inference. Under certain conditions, functions of complete sufficient statistics, such as the ratio of a random variable to the sum of random variables, can be shown to be independent of the complementary statistic. This independence property is useful for deriving distributions of transformed variables and for simplifying the analysis of complex statistical models.

Sufficient Statistic

A sufficient statistic for a parameter is a function of the data that contains all the information needed to estimate that parameter. The factorization theorem provides a formal way to identify sufficient statistics by expressing the joint density as a product of a function that depends on the data only through the statistic and another function that does not depend on the parameter. Using sufficient statistics simplifies the estimation process by reducing the data while retaining all relevant information.

Unbiasedness of Estimators

An estimator is said to be unbiased if its expected value equals the true value of the parameter it is estimating. This property means that on average, across many samples, the estimator neither overestimates nor underestimates the parameter. Checking unbiasedness involves computing the expectation of the estimator and verifying that it equals the parameter of interest.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which represents the probability of observing the given sample. It involves setting up the joint probability density or mass function for the sample and then finding the parameter value(s) that maximize this function. The MLE has desirable properties such as consistency and asymptotic normality under regularity conditions.

Completeness in Statistics

A statistic is complete if there are no nontrivial functions of the statistic that have an expected value of zero for all values of the parameter. Completeness is an important property because, together with sufficiency, it guarantees the uniqueness of any unbiased estimator that is a function of that statistic. This concept is pivotal in proving the optimality of estimators through results such as the Lehmann–Scheffé theorem.

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